Musical Actions of Dihedral Groups

TL;DR

This paper explores the dual actions of the dihedral group of order 24 on musical melodies and chords, providing geometric and algebraic insights that enhance analysis of diverse musical works.

Contribution

It reveals the duality of two dihedral group actions on melodies and chords, connecting algebraic and geometric perspectives in music theory.

Findings

Both actions are geometrically and algebraically dual.

The duality has been applied to analyze works from Hindemith to the Beatles.

The paper clarifies the mathematical structure underlying musical transformations.

Abstract

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.